Project Summary Many genome-wide association studies (GWASs) have been published on complex diseases, where data on a large number of single nucleotide polymorphisms (SNPs) are collected to study the association between these SNPs and a disease. Although new loci are found to be associated, together they generally explain very little of the additive genetic variation (heritability) estimated from biometric modeling of these traits in families. Recently a series of papers1,2 demonstrated that much of the heritability is not `missing' but is `hidden', unable to be detected with existing sample sizes. They estimated the total variance explained by the genome-wide SNPs (SNP-based heritability) using the correlation among the individuals from population-based unrelated samples. These methods are now being widely used to determine SNP-based heritability of many phenotypes.3,4,5 How- ever, several recent studies have questioned the accuracy of this heritability estimation.6,7,8 Moreover, the in?uence of genes on these diseases may also vary over time through interaction with factors such as age, culture or on other location and time-dependent environmental factors. This proposal responds to the need by developing advanced statistical methods and ef?cient computing algorithms to analyze GWAS data for robust estimation of SNP-based heritability. We propose to develop computationally ef?cient methods to investigate heritability in a cohort study by developing a dimension reduction approach motivated by the Gaussian predic- tive process model.9 We aim to utilize the spatial dependency in the genetic relationships among the sampled individuals to dictate this dimension reduction. The proposed methods are motivated by and will be applied to the data from Minnesota Center for Twin and Family Research (MCTFR), a longitudinal genome-wide study on genes and environments and their interactions with different behavioral traits. We intend to study SNP-based heritability for different substance use disorders (SUDs). Similar phenotypes considered in the MCTFR study will be used to derive SNP-based heritability estimates in the UK Biobank cohort data. Open access user-friendly statistical software will be developed and distributed.